A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
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Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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18 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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Transient discrete time Markov chain on integers: can direction of flow be proven?

I'm not very familiar with the theory of Markov chains, and I'd like to learn how complicated the following problem actually is. Let there be a discrete time Markov chain on $\mathbb{Z}$, where the ...
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1answer
36 views

Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
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Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
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18 views

Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
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1answer
22 views

Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
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1answer
32 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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1answer
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Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
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2answers
34 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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11 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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20 views

Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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Aggregate arrivals from a renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process". The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF $f(t)$. ...
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1answer
25 views

Aggregate arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=λe^{−λt},~t>0$. ($λ$ is the arrival rate of the ...
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29 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
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34 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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14 views

process stochastics and branching process [duplicate]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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1answer
27 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
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discontinuous Gaussian field

I am trying to build an example of a discontinuous Gaussian field. The simplest I could come up with is the following: Let $Y,Z$ be two independent brownian motions on $[0,1]$, and $T$ a uniform ...
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21 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
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1answer
57 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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0answers
33 views

$\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$

Let $(B_t)_{t \ge 0}$ and $(C_t)_{t \ge 0}$ be two standardized brownian motions. Now why is $\sup_{t \ge 0} B_t$ distributed same as $\sup_{t \ge 0} C_t$? This is a result we assumed as trivial ...
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1answer
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$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
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23 views

FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
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24 views

Simple Markov property

I want to prove the simple Markov property but I come to a point where I do not see how to conclude. I want to prove $\mathbb{E}_\nu[Z\circ\Theta_t\mid ...
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25 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
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Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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19 views

Represent stochastic process as conditional expectation

I try to reduce my problem to the following question, which is stated rather sloppy (without possibly necessary additional conditions). Let $Y_t$ be a real stochastic process for $t \in [0, T]$ and ...
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1answer
138 views

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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23 views

Prove that an integral is zero (from Gardiner's Handbook of stochastic methods)

I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral $\sum_i\int d\vec x \frac{\partial}{\partial ...
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1answer
27 views

Central limit theorem - generalizations [on hold]

I am looking for some generalizations for the Central limit theorem in the following sense: Let $\phi$ be a function on the natural numbers, under what conditions on $\phi$ $ ...
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1answer
21 views

construct a martingale process from any process [closed]

If ${Z_n, n \geq 0}$ is any sequence of integrable random variables, then ${\sum_{i=1}^{n}[Z_i-E(Z_i|Z_{i-1},...,Z_1)]}$ is a martingale relative to the sequence of $\sigma$-fields generated by $Z_i$, ...
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1answer
28 views

How to get the basis of $L^2[0,1]$ from the basis of $L^2[0,2]$

Is there any way to derive orthonormal basis of $L^2[0,1]$ from the orthonormal basis of $L^2[0,2]$? Here $L^2[0,2]$: is space of square integrable centered stochastic process on $\Omega\times[0,2]$, ...
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about martingale

The definition about martingale process is $E(Z_{n+1}\mid \mathcal F(X_n))=Z_n$, where $\mathcal F(X_n)$ is the $\sigma$ field generated by $X_n$. My question is if $E(Z_{n+1}\mid \mathcal F(X_n) ) = ...
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1answer
32 views

Why use stopping times rather than a deterministic sequence to localise a martingale?

I am a beginner on stochastic processes I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the ...
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How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
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Ito Isometry on Multivariable indicator function

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The HODRICK-PRESCOTT Filter For The Frequency of Observations" I will very much appreciate ...
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1answer
26 views

Calculating $ \mathbb E \left[e^{-\mu W_T } 1_\left( {\min W_t \leq a} \right) \right]$ for a Wiener process

Let $W_t$ be a standard Wiener process, $a$ some real number, and $\chi (x)$ the indicator function. I am trying to calculate the following expectation: $$ \mathbb E \left[e^{-\mu W_T } \chi \left( ...
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0answers
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Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
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1answer
26 views

“The well-known formulas that gives the relation between the generating functions of a sequence and the sequence of its 'tails'”

I'm reading a paper on Branching Processes and the Theory of Epidemics, and the fourth page (p. 262 of the book) the author refers to "the well-known formulas that gives the relation between the ...
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1answer
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Stationary Process, shift-invariant, translation invariant? [closed]

If I have a stationary stochastic process... can i also say it is translation-invariant or shift-invariant? Does this all mean the same?
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1answer
36 views

$ P(W_t - W_\tau > 0 \text{ and } \tau <t) = \frac{1}{2}P(\tau < t) $ for a stopping time $\tau$

Let $W_t$ be a standard Wiener process and $\tau = \min \lbrace t \geq 0 :W_t \geq a \rbrace$, the first time the process reaches level $a$. By symmetry of the Gaussian distribution we have $$ P(W_t ...
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0answers
12 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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4answers
84 views

Mathematical philosophical questions about the general theory of stochastic processes.

After 6 months spent on what is termed the "general theory" of stochastic processes and after having worked out many nuances of the field, I realized that: The general theory is beautiful ...
2
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1answer
49 views

Compound Poisson Process function expected value

For the calculus of a financial derivatives, I need to compute the next expectation: $$\mathbb{E}\left((\sum_{i=1}^{N_T} (J_i-k))_+\mid J_1+\cdots + J_{N_t}=x \right)$$ where $$(X_t-k)_+= ...
1
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0answers
44 views

Conditional expectation of stochastic integral with independent components

Let $T$ denote a maturity and $\mathbb{F}$ a filtration. Besides, consider two processes $A$ and $B$ which are mutually independent and are both dependent on (a subfiltration of) $\mathbb{F}$. Does it ...